{"id":"614ba56d-65a2-4414-9d4f-ce8d94a76c20","arxiv_id":"2608.10285","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Any finite modular tensor category gives a 3-dimensional derived TQFT whose state spaces are derived Hom spaces, recovering known non-semisimple TQFTs after taking zeroth cohomology.","lead":"This paper builds 3-dimensional topological quantum field theories from derived representation categories of quantum groups and other finite modular tensor categories. It connects a new derived-level construction to known non-semisimple TQFTs, and proposes these as mathematical versions of certain twisted supersymmetric field theories.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"State-space identification in Proposition 10.5 is imported from [22,20] and its reduction functor p is asserted rather than proved; the derived state spaces and H^0 comparison collapse if this fails.","rationale":"The reader identified Proposition 10.5 as the weakest assumption, and I agree. The central theorem's two headline outputs--the derived state-space identification RHom_A(C^{⊗g},1) and the H^0 recovery of the abelian De Renzi-Gainutdinov-Geer-Patureau-Mirand-Runkel theory--both flow from the natural identification of L_Afin on marked surfaces with Hom^*_Afin(C^{⊗g}, m_f -). That identification is not proved in this paper; it is cited from [20, Theorem 8.5] and [22, Proposition 4.17], and the reduction functor p that transports bordisms from the marked category into the framework of [20] is asserted to be well-defined by relations U1 and U3 without a detailed check. This is not a disagreement with the consensus that those cited results are correct; it is a correctness risk in the chain of dependencies. If the cited statement is exactly as strong as needed, the construction should go through. If it is only proved for a single positive marking, or if the reduction functor is not a genuine functor, then the derived state spaces and the comparison with the abelian theory fail at the first step, before any of the ∞-categorical localization or Kan extension machinery is used. The paper's own remarks (1.7, 1.8) show that the authors are careful to flag unproved strengthenings, which makes the absence of a proof or precise quotation for this base input all the more notable. A conditional verdict is therefore appropriate: the main construction is plausible and internally coherent, but it rests on an imported theorem whose precise scope should be checked before full acceptance.","tokens_in":62812,"tokens_out":11947,"duration_ms":138162,"concrete_test":"Independently verify the imported input: for a concrete non-semisimple finite modular tensor category (e.g. Rep_q(SL2) for q a primitive 2p-th root of unity with p > 2), compute both sides of Proposition 10.5 for a genus-1 surface with a single projective marking P and with a non-projective simple marking, using the explicit formulas in [22,20], and check the natural isomorphism is compatible with composition of two merging bordisms. In particular, re-derive p: hBord_Afin -> Bord_abelian on a two-step composition of marked bordisms and check L_Afin(M_2 ∘ M_1) equals L_Afin(M_2) ∘ L_Afin(M_1) on the identified state spaces. If the identification holds for all indecomposable projectives and simples, the derivation chain is secure; if it fails for a projective marking, the central claim must be weakened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is the abelian LRT input. Theorem 10.1 constructs L_Afin by composing a reduction p: hBord_Afin -> Bord_abelian with Z* from [20, Theorem 9.3]; the proof asserts p is well-defined via relations [20, U1, U3] and composition-compatible, but gives no verification. Proposition 10.5 then uses L_Afin together with Lemma 10.4 to identify L_Afin(Σ_x) with Hom^*_Afin(C^{⊗g}, m_f x) for all marked connected surfaces. This natural isomorphism is the base for all later state-space calculations: Lemma 10.9 extends it to unbounded cochains, Proposition 13.5 lifts it to Maps_K, Section 15 lifts it to RHom_A, Proposition 15.8 compares H^0 L_Dfin with L_abelian, and Theorem 17.10's unmarked state spaces RHom_A(C^{⊗g},1) depend on it. If the reduction p fails to be a functor, or Proposition 10.5 is false for some marking x (e.g. a projective marking), then the derived state-space identification and the H^0 recovery claim fail even though the ∞-categorical machinery is formally sound. The paper's own Remarks 1.7 and 1.8 concede that some advertised strengthenings are not proved, but those remarks do not cover this imported base.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs, for any finite modular tensor category A, a symmetric monoidal functor from a marked non-compact 3-dimensional bordism ∞-category to the ∞-category of dg vector spaces, with state spaces identified with linearized mapping spaces of the derived category D^b(A_fin). It then claims that after a homotopy truncation all markings can be removed, yielding an unmarked TQFT Bord^nc_{3,2} -> D(Vect) whose genus-g state spaces are RHom_A(C^{⊗g}, 1), whose H^0 recovers the abelian non-semisimple LRT theory of De Renzi et al., and which produces projective mapping class group actions on Hochschild cohomology as well as power-series valued 3-manifold and knot invariants. The construction proceeds by importing the abelian LRT theory from [22,20], localizing the resulting theory along homotopy equivalences, applying a relative Kan extension to the derived ∞-category, and then localizing further along a class of bordisms with universal skeins to erase markings.","tokens_in":63091,"tokens_out":7966,"duration_ms":81481,"significance":"If the proof obligations flagged below are met, this would be a substantial contribution to non-semisimple TQFT: it provides an explicit homotopical/derived realization of Lyubashenko-Reshetikhin-Turaev theory, identifies state spaces directly with derived mapping spaces, and gives a conceptual mechanism for deriving the projective mapping class group actions on Hochschild cohomology observed in [51,72]. The overall architecture—vertical localization, relative Kan extension, then a further localization in the homotopy category—is coherent and well motivated, and the paper is honest in Remarks 1.7 and 1.8 about which advertised strengthenings are not proved. The main concern is that several load-bearing steps are imported or stated without the proofs needed to verify the central claims.","major_comments":[{"comment":"The construction of L_{A_fin} rests on two assertions that are not verified in the text. Theorem 10.1 defines the reduction functor p: hBord^nc_{A_fin} -> Bord^nc_{A_fin} and states that well-definedness follows from relations [20, U1, U3] and composition compatibility from [20, U1], but no proof or precise quotation of these relations is given. Proposition 10.5 then imports the state-space identification L_{A_fin}(Σ_-) ≅ Hom^*_{A_fin}(C^{⊗g}, m_f -) from [22, Prop. 4.17] and [20, Thm. 8.5]. This identification is the base for Lemma 10.9, Proposition 13.5, the derived state-space formulas in §15, Proposition 15.8, and Theorem 17.10. If p fails to be functorial, or if Proposition 10.5 fails for some marking type (for instance a projective marking), the derived state spaces and the H^0 comparison with [22] collapse. Please provide a proof, or a complete statement of the imported theorem with its exact hypotheses, and specify whether the cited results cover arbitrary markings or only admissible ones.","section":"§10.1–10.3, Theorem 10.1 and Proposition 10.5"},{"comment":"In the version supplied for review, the derivations leading to the two main theorems are not present as proofs: §15 states Theorem 15.1/15.5 after a Kan-extension argument, and §16–17 state the localization at Θ_D and the unmarking theorem 17.10, including the claimed equivalence hBord^nc_*[Θ_*^{-1}] ≃ Bord^nc_{3,2}, but no verification is included. Since Theorem 17.10 is exactly the step that removes all markings and produces the unmarked TQFT with state spaces RHom_A(C^{⊗g},1), and since the projective mapping class group actions of Corollary 17.13 depend on it, the central advertised result cannot be checked from the submitted material. The full proof of the unmarking equivalence and of the localization construction should be included, or the claims should be explicitly downgraded to conjectures.","section":"§15–17, Theorem 15.1/15.5 and Theorem 17.10"},{"comment":"Equation (5) defines the invariant Inv(A|\\check M) using determinants of H^n L_{D_fin}(\\check M); however Remark 1.1 explicitly allows infinite-dimensional state spaces in the non-compact setting. The paper should either prove that each H^n is finite-dimensional in the cases used for Propositions 19.4 and 19.6, or explain how the determinant is to be understood for infinite-dimensional endomorphisms. Without this, the power-series invariants in dimension 3 are not well-defined.","section":"§1.5, Eq. (5), and Remark 1.1"}],"minor_comments":[{"comment":"The notation switch from A in the introduction to A^♡ in later sections should be announced before it appears in displayed theorems; a small notation table would avoid confusion.","section":"§1.8 and Theorem statements"},{"comment":"The composition of mapping complexes by gluing cylindered bordisms is described informally; an associativity/coherence verification or a precise reference to a standard construction would be helpful.","section":"§8.3"},{"comment":"There are several typos: 'closed 3-manifold' in the abstract should be 'closed 3-manifolds', and 'dimensiona 4' in Section 2.1 should read 'dimension 4'.","section":"Abstract and §2.1"},{"comment":"The unproved expectations, such as agreement of the mapping class group actions with [51,72] and the conjectural ∞-categorical unmarked lift, should be collected in a clearly delimited 'scope and conjectures' subsection so that they are not mistaken for proved results.","section":"Remarks 1.7 and 1.8"}],"recommendation":"major_revision","confidential_remarks":"The proposed framework is promising and the architecture is coherent, but the review copy omits proofs of the two most central steps: the Kan-extension construction in §15 and the unmarking theorem in §16–17. I would ask the authors to supply complete proofs of these sections and to state Proposition 10.5 as an imported theorem with its exact hypotheses before a second round of review."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read this paper as a serious, well-structured contribution. It does what the abstract promises: for any finite modular tensor category A, it builds a symmetric monoidal functor from an infinity-category of marked bordisms to dg vector spaces, identifies the state spaces on a genus-g surface with Maps_{D(A)}(C^{⊗g}, m_I(x)), and then shows that a homotopy truncation removes the markings, producing an unmarked TQFT with state spaces RHom_A(C^{⊗g}, 1). The H^0 comparison with the De Renzi–Gainutdinov–Geer–Patureau-Mirand–Runkel abelian theory is a nice checkpoint, and the derived state spaces are a genuinely new way to organize a lot of non-semisimple phenomena. The paper is also honest: Remarks 1.7 and 1.8 explicitly say that some advertised connections are not proved, and the conjectures in Section 18 are clearly labeled as such.\n\nThe soft spot is exactly what the stress-test note flags. Proposition 10.5, the state-space identification for the abelian theory, is imported from [20] and [22]. The reduction functor p in Theorem 10.1 is asserted rather than checked in detail; the proof says it follows from relations U1 and U3, but does not show the verification. That is the base for everything that follows: Lemma 10.9, Proposition 13.5, Section 15, and the H^0 recovery all rest on it. If that functor is not actually well-defined, or if Proposition 10.5 fails for some marking, the derived theory does not collapse formally---the infinity-categorical machinery could still stand---but the state-space identifications and the comparison with the abelian theory would not follow. This is not circularity and not fitting; it is a straightforward dependency on prior work. My read is that the dependency is real but the prior work is credible, and the paper is transparent about what it cites rather than reproves.\n\nFor a referee, I would want the check to focus on Theorem 10.1 and Proposition 10.5: verify that p respects composition and passes through the stated relations, and that the natural isomorphism in Proposition 10.5 is genuinely natural in the markings. The unmarking sections (16–17) are the other place worth a closer look, since the reviewer had only partial visibility there. I do not see a fatal flaw, and the central argument holds up as far as I can tell.\n\nThis paper deserves a serious referee. It is long but the writing is unusually honest about what is proved, what is imported, and what is conjectural. I would cite it if I were working in non-semisimple TQFTs, and I would bring it to a reading group for the architectural ideas even if not for every technical detail.","headline":"A substantial and plausible construction of derived 3d TQFTs for finite modular tensor categories; the main load-bearing input is imported abelian LRT theory, and that input deserves careful checking before the rest is trusted.","tokens_in":63612,"tokens_out":1666,"would_cite":true,"duration_ms":20438,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["18M15","81T45","17B37","16E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper constructs a 3-dimensional topological field theory from the derived $\\infty$-category of any finite modular tensor category, with genus-$g$ state spaces given by derived Homs into powers of the canonical coend, and degree-zero…","keywords":["topological quantum field theory","derived ∞-category","modular tensor category","quantum group representations","Reshetikhin–Turaev theory","Hochschild cohomology","mapping class group","∞-categories"],"falsifier":"Take $A=(\\mathrm{Rep}_q SL_2)_{\\mathrm{small}}$ at a root of unity and compute the degree-zero cohomology of $\\mathrm{RHom}_A(C^{\\otimes 2},\\mathbf{1})$ for a genus-2 surface, comparing its dimension with the known state-space dimension of the abelian non-semisimple theory on the same surface; a mismatch of dimensions would falsify the claimed $H^0$ identification. Alternatively, compute the $SL_2(\\mathbb{Z})$ matrices acting on $HH^*(A)$ via the unmarked theory on the torus and compare them with the independently known genus-one action, so that disagreement in any one matrix entry would falsify the projective-action claim.","tokens_in":62600,"feed_emoji":"🌀","tokens_out":15854,"duration_ms":125137,"temperature":0.7,"pith_summary":"Non-semisimple 3-dimensional TQFTs have until now been built at the abelian level, with projective objects doing specialised work; this paper argues they are shadows of a derived theory. For any finite modular tensor category $A$, it constructs a symmetric monoidal functor from a bordism $\\infty$-category of surfaces and 3-manifolds marked by objects of the derived $\\infty$-category $D_{\\mathrm{fin}}=D^b(A_{\\mathrm{fin}})$ to the $\\infty$-category of dg vector spaces. The state space of a connected genus-$g$ surface is identified with the derived Hom space $\\mathrm{RHom}_A(C^{\\otimes g},\\mathbf{1})$, where $C$ is the canonical coend, so the torus recovers Hochschild cohomology and the sphere recovers extensions of the unit. Removing the markings by homotopy truncation yields an unmarked theory, and its degree-zero cohomology reproduces the abelian non-semisimple Reshetikhin–Turaev theory. The payoff is a single 3-dimensional home for projective mapping class group actions on cohomology, power-series valued knot and 3-manifold invariants, and a candidate mathematical realisation of derived A-twisted supersymmetric theories.","feed_headline":"Derived categories yield 3D TQFTs from modular tensor categories","feed_subtitle":"State spaces are derived Homs into powers of the canonical coend; degree zero recovers the non-semisimple Reshetikhin–Turaev theory.","key_machinery":"The load-bearing object is the derived $\\infty$-category $D_{\\mathrm{fin}}=D^b(A_{\\mathrm{fin}})$ of a finite modular tensor category $A$, viewed as a framed-disk ($f rDisk$) monoidal $\\infty$-category via the duality equivalence between $A^{\\mathrm{op}}$ and pro-finite complexes. From any such framed-disk monoidal $\\infty$-category one builds a marked bordism $\\infty$-category $\\mathrm{Bord}^{\\mathrm{nc}}_E$: surfaces carry disks labelled by objects, 3-bordisms carry embedded cylinders labelled by morphisms, and homotopies of these cylinders encode skein-like relations. The TQFT is then produced in three moves: localisation along homotopy equivalences yields the homotopy $\\infty$-category version, relative Kan extension carries it to the derived $\\infty$-category, and a final localisation with homotopy truncation removes the markings. The identity doing the state-space computation is the canonical coend $C=m_R(\\mathbf{1})$, defined as the image of the unit under the right adjoint to the multiplication functor; powers $C^{\\otimes g}$ feed into the derived Homs $\\mathrm{RHom}_A(C^{\\otimes g},\\mathbf{1})$, making the torus value exactly Hochschild cohomology.","core_discovery":"The paper's central claim is that a finite modular tensor category $A$ determines a 3-dimensional derived TQFT through its derived $\\infty$-category of bounded finite-length complexes. Concretely, there is a symmetric monoidal functor $L_{D_{\\mathrm{fin}}}\\colon \\mathrm{Bord}^{\\mathrm{nc}}_{D_{\\mathrm{fin}}}\\to \\mathrm{Vect}$, and for every connected genus-$g$ surface with marking tuple $x$ the state space is naturally $\\mathrm{Maps}_{D_{\\mathrm{fin}}}(C^{\\otimes g}, m_I(x))$, the linearized mapping space of the derived category. Restricting to unit labels and applying homotopy truncation produces an unmarked theory $L_{D_{\\mathrm{fin}}}\\colon \\mathrm{Bord}^{\\mathrm{nc}}_{3,2}\\to D(\\mathrm{Vect})$ whose genus-$g$ state space is $\\mathrm{RHom}_A(C^{\\otimes g},\\mathbf{1})$; the degree-zero part of this theory is the non-compact part of the abelian non-semisimple LRT theory. When $A$ is semisimple, the construction collapses to the non-compact part of classical Reshetikhin–Turaev theory. The paper expects, but does not verify, that the induced projective $SL_2(\\mathbb{Z})$ action on Hochschild cohomology coincides with previously defined actions.","pith_inferences":["If the conjectured unmarked $\\infty$-categorical lift exists, the projective mapping class group representations would be the 2-dimensional shadow of a genuine 3-dimensional TQFT, suggesting a fully extended theory whose value on $S^1$ is the derived category itself.","The explicit state-space formula makes a concrete numerical check available outside the paper: for $A=(\\mathrm{Rep}_q SL_2)_{\\mathrm{small}}$, the unknot invariant in genus 1 should have nontrivial $t$-adic higher coefficients governed by $HH^{>0}(A)$, testing whether derived information truly leaks into knot invariants.","The same localize-then-Kan-extend template could produce derived lifts of other projective-dependent TQFTs, such as homology-dependent unrolled theories and non-semisimple state-sum theories, since the construction only needs a preceding abelian theory for a marking scheme.","The conjectural deformation along local systems suggests the derived theory is the trivial-local-system stalk of a relative TQFT over the moduli of flat connections; a testable precursor would be computing derived skein modules of the solid torus and comparing them with functions on local systems."],"forward_implications":["For genus one, the unmarked theory makes $\\mathrm{RHom}_A(C,\\mathbf{1})$, identified with the derived Hochschild cohomology of $A$, carry a projective $SL_2(\\mathbb{Z})$-action, packaging the known genus-one actions as part of a 3-dimensional field theory.","For a connected genus-$g$ surface, every mapping class acts projectively on $\\mathrm{RHom}_A(C^{\\otimes g},\\mathbf{1})$, giving dg representations of the central extension of the mapping class group.","When $A$ is semisimple, the derived theory reduces to the non-compact part of the classical Reshetikhin–Turaev theory, and the full theory is its unique extension to closed bordisms.","In dimension 3, the theory produces power-series valued invariants of framed knots (genus 1) and of closed 3-manifolds (genus 0); the 3-manifold series is determined by its constant term from the abelian theory, whereas knot series can involve higher Hochschild cohomology.","Taking degree-zero cohomology recovers the marked and unmarked non-compact parts of the abelian non-semisimple LRT theory, so the new theory is a genuine derived lift rather than a separate invariant."],"supporting_citations":[{"why":"Supplies the abelian non-semisimple LRT TQFT whose marked and unmarked non-compact parts the derived theory recovers at degree zero.","marker":"[22]"},{"why":"Supplies the natural state-space identification on marked finite surfaces, quoted as Proposition 10.5, that anchors the derived state-space computation.","marker":"[20]"},{"why":"Supplies the localization framework for monoidal $\\infty$-categories used in the vertical localization step from chains to homotopy categories.","marker":"[43]"},{"why":"Supplies the $\\infty$-operad and Kan-extension framework used to move from the homotopy to the derived bordism category with monoidal structure.","marker":"[57]"},{"why":"Gives the projective mapping class group actions on cohomology that the genus-one action is expected to recover.","marker":"[51]"},{"why":"Provides the genus-one mapping class group and Hochschild cohomology action context that the unmarked theory is designed to match.","marker":"[72]"}],"fun_headline_variants":["Derived 3D TQFT from modular tensor categories","Finite modular categories yield derived TQFTs in 3D","Derived categories of quantum group reps give 3D TQFTs","Mapping spaces define derived 3D TQFTs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction leans on the imported identification, stated as Proposition 10.5, that the abelian theory's state space on any marked connected surface is naturally isomorphic to $\\mathrm{Hom}^*_{A_{\\mathrm{fin}}}(C^{\\otimes g}, m_f -)$ and that this is functorial for merging bordisms; if that input fails, the derived state-space formula and the $H^0$ comparison collapse.","fun_headline_variants_meta":{"raw":{"variants":["Derived 3D TQFT from modular tensor categories","Finite modular categories yield derived TQFTs in 3D","Derived categories of quantum group reps give 3D TQFTs","Mapping spaces define derived 3D TQFTs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000285,"raw_usage":{"total_tokens":1781,"prompt_tokens":1153,"completion_tokens":628,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":769,"completion_tokens_details":{"reasoning_tokens":553}},"tokens_in":769,"tokens_out":628,"duration_ms":5772,"temperature":1.0,"reasoning_tokens":553,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:10:36.949690+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take $A=(\\mathrm{Rep}_q SL_2)_{\\mathrm{small}}$ at a root of unity and compute the degree-zero cohomology of $\\mathrm{RHom}_A(C^{\\otimes 2},\\mathbf{1})$ for a genus-2 surface, comparing its dimension with the known state-space dimension of the abelian non-semisimple theory on the same surface; a mismatch of dimensions would falsify the claimed $H^0$ identification. Alternatively, compute the $SL_2(\\mathbb{Z})$ matrices acting on $HH^*(A)$ via the unmarked theory on the torus and compare them with the independently known genus-one action, so that disagreement in any one matrix entry would falsify the projective-action claim.","supporting_citations":[{"cited_title":"De Renzi, A","cited_arxiv_id":null,"evidence_quote":"Supplies the abelian non-semisimple LRT TQFT whose marked and unmarked non-compact parts the derived theory recovers at degree zero."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the localization framework for monoidal $\\infty$-categories used in the vertical localization step from chains to homotopy categories."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the $\\infty$-operad and Kan-extension framework used to move from the homotopy to the derived bordism category with monoidal structure."},{"cited_title":"Lentner, S","cited_arxiv_id":null,"evidence_quote":"Gives the projective mapping class group actions on cohomology that the genus-one action is expected to recover."},{"cited_title":"Schweigert and L","cited_arxiv_id":null,"evidence_quote":"Provides the genus-one mapping class group and Hochschild cohomology action context that the unmarked theory is designed to match."}],"review_version":1}